ORBIT GROWTH of CONTACT STRUCTURES AFTER SURGERY Patrick Foulon, Boris Hasselblatt, Anne Vaugon

ORBIT GROWTH OF CONTACT STRUCTURES AFTER SURGERY Patrick Foulon, Boris Hasselblatt, Anne Vaugon

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Patrick Foulon, Boris Hasselblatt, Anne Vaugon. ORBIT GROWTH OF CONTACT STRUCTURES AFTER SURGERY. 2019. hal-02336690

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PATRICK FOULON, BORIS HASSELBLATT, AND ANNE VAUGON

Abstract. Investigation of the effects of a contact surgery construction and of invariance of contact homology reveals a rich new field of inquiry at the intersection of dynamical systems and contact geometry. We produce contact 3-flows not topologically orbit-equivalent to any algebraic flow, including examples on many hyperbolic 3-manifolds, and we show how the surgery produces dynamical complexity for any Reeb flow compatible with the resulting contact structure. This includes exponential complexity when neither the surgered flow nor the surgered manifold are hyperbolic. We also demonstrate the use in dynamics of contact homology, a powerful tool in contact geometry.

Contents 1. Introduction 1 2. The results 3 2.1. Definitions and notations 3 2.2. New contact flows 4 2.3. Production of closed orbits for contact Anosov flows 6 2.4. Production of closed orbits for any Reeb flow 7 3. Surgery and production of closed orbits 11 3.1. The surgery from the contact viewpoint 11 3.2. Surgery on unit tangent bundle and Anosov flows 12 3.3. Impact on entropy 15 4. Contact homology and its growth rate 16 5. Orbit growth in a free homotopy class for degenerate contact forms 19 6. Exponential growth of periodic orbits after surgery on a simple geodesic 21 7. Coexistence of diverse contact flows—proof of theorem 2.23 23 7.1. Dynamical properties of the periodic Reeb flow after surgery 23 7.2. Proof of theorem 2.23 24 References 27

1. Introduction This paper is a sequel of [29] in which the authors decribed a surgery construction adapted to contact flows. This construction was originally conceived as a source of uniformly hyperbolic contact flows. However it turns out that the surgered flows exhibit more noteworthy dynamical properties than orginally observed and that interesting consequences of the surgery arise even when the initial or resulting flow are not hyperbolic. Thus the primary interest in this contact surgery may be as a

Key words and phrases. Anosov flow, 3-manifold, contact structure, Reeb flow, surgery, contact homology. Partially supported by the Committee on Faculty Research Awards of Tufts University. Partially supported by ANR QUANTACT. 1 2 PATRICK FOULON, BORIS HASSELBLATT, AND ANNE VAUGON rich source of contact flows exhibiting new phenomena from both dynamical and contact points of view. The starting point of our surgery is the unit tangent bundle M of a surface of negative and (mainly, but not always necessarily) constant curvature equipped with its natural contact structures. This surgery is known to contact-symplectic topologists as a Weinstein surgery, and its desciption in [29] makes it easier to study dynamical properties (as opposed, for instance, to topological properties). Our purpose is to expand the understanding of the dynamical effects achieved by the contact surgery from [29] in 3 main directions: we show that the complexity of the resulting flow exceeds that of the flow • on which the surgery is performed (Theorems 2.10, 2.12, 2.18, and 2.23), we show that much of the complexity of the resulting flow is reflected in • the cylindrical contact homology and is therefore realized in any Reeb flow associated to the contact structure resulting from the surgery (Theorems 2.18, 2.22, and 2.23), and we do this beyond the context of hyperbolic flows (in more than one way— • Theorems 2.22 and 2.23). Taken together, this reveals a much richer field of inquiry at the interface between contact geometry and dynamical systems than was apparent when the surgery construction was conceived. Contact homology and its growth rate are relevant tools to describe dynamical properties of all Reeb flows associated to a given contact structure. Even if it is not always explicit in the statements, they play a crucial role in the proofs of Theorems 2.18, 2.22, and 2.23. A goal of this paper is to demonstrate to dynamicists the use of these powerful tools from contact geometry. In addition to the dynamical point of view, our study is also motivated by contact geometry as we want to investigate connections between growth properties in Reeb dynamics (generally characterized by the growth rate of contact homology) and the geometry of the underlying manifold. The simplest model of such a connection is Colin and Honda’s conjecture [18, Conjecture 2.10], and some surgeries under study give examples supporting it. Colin and Honda speculate that the number of Reeb periodic orbits of universally tight1 contact structures on hyperbolic manifolds grows at least exponentially with the period. More generally, one may look for sources of exponential or polynomial behavior of contact homology. Our starting point, the unit tangent bundle of an hyperbolic surface, is a transitional example as it carries two special contact structures, one with an exponential growth rate for contact homology and one with a polynomial growth rate. We prove (Theorems 2.22 and 2.23) that some surgeries lead to two coexisting contact forms on the surgered manifold with exponential and polynomial growth rates and therefore give new examples of transitional manifolds with respect to growth rate. Note that these examples do not include hyperbolic manifolds (and are therefore compatible with Colin and Honda’s conjecture). The principal results in this article were obtained in 2014, and we here integrate it with work by others that was done contemporaneously [1, 2, 3].

Structure of the paper. In section 2 we cover the background material and present our main results. Specifically, section 2.2 presents and elaborates our earlier results [29], and section 2.3 introduces the resulting complexity increase of the surgered geodesic flow. section 2.4 finally describes how cylindrical contact homology forces complexity of Reeb flows with the same contact structure, introduces our surgery

1see section 2.1 ORBIT GROWTH OF CONTACT STRUCTURES AFTER SURGERY 3 on the fiber flow, and discusses the relation of our results to other works on contact surgery and Reeb dynamics. The construction of contact surgery is recalled in section 3, which also contains some preliminary results on the dynamics of the surgered flow and the proof of theorem 2.10. In section 4 we define contact homology and its growth rate. This enables us to prove theorem 2.18 in section 5, theorem 2.22 in section 6 and theorem 2.23 in section 7.

Acknowledgements. We thank Marcelo Alves, FrédéricBourgeois, Patrick Massot and Samuel Tapie for useful discussions and helpful advice. Boris Hasselblatt is grateful for the support of the ETH, which was important as we finalized this work.

2. The results 2.1. Definitions and notations. A manifold is said to be closed if it is compact and has no boundary. A C∞ 1-form α on a 3-manifold M is called a contact form if α dα is a volume form. The associated plane field ξ := ker α is a cooriented contact∧ structure, and (M, ξ) is called a contact manifold.. The geometric object under study in contact geometry is the contact structure (as opposed to the contact form). Note that for a given contact structure ξ = ker(α) the contact forms with kernel ξ are exactly the forms fα where f ∞(M, R r 0 ). Additionally, if α dα is a volume form then ∈ C { } ∧ fα d(fα) is also a volume form for any f ∞(M, R r 0 ). A curve tangent to ξ is∧ said to be Legendrian. ∈ C { } The Reeb vector field associated to a contact form α is the vector field Rα such that ιRα α = 1 and ιRα dα = 0. Its flow is called the Reeb flow (and it preserves α because Rα α = ιRα d α = 0). Note that the Reeb vector field is associated to a contactL form α: if we consider another contact form α0 = fα where ∞ 0 0 f (M, R r 0 ), then dα = df α + f dα and the condition ιR 0 dα = 0 ∈ C { } ∧ α implies that Rα and Rα0 are not collinear unless f is constant. A Reeb field on a contact manifold (M, ξ) is the Reeb field of any contact form α with ξ = ker α. By Libermann’s Theorem [38] on contact Hamiltonians (see for instance [30, Theorem 2.3.1]), these are exactly the nowhere-vanishing vector fields transverse to ξ whose flows preserve ξ. A Reeb vector field (or the associated contact form) is said to be nondegenerate if all periodic orbits are nondegenerate (1 is not an eigenvalue of the differential of the Poincarémap). A Reeb vector field (or the associated contact form or the associated contact structure) is said to be hypertight if there is no contractible periodic Reeb orbit. One can always perturb a contact form into a nondegenerate contact form. Hypertightness is much more restrictive. Contact structures on 3-manifolds can be divided into two classes: tight contact structures and overtwisted contact structures. This fundamental distinction is due to Eliashberg [21] following Bennequin [11]. Tight contact structures are the contact structures that reflect the geometry of the manifolds and this article focuses on them. A contact structure ξ is said to be overtwisted if there exists an embedded disk tangent to ξ on its boundary. Otherwise ξ is said to be tight. Universally tight contact structures are those with a tight lift to the universal cover. Universally tight and hypertight [33] contact structures are always tight. All the contact structures considered in this paper are hypertight and therefore tight. We recall from [29] a contact surgery on a Legendrian curve γ SΣ derived from a closed geodesic c on a hyperbolic surface Σ. This corresponds∈ to a (1, q) Dehn- − surgery and results in a new manifold MS with a contact form αA. The construction 4 PATRICK FOULON, BORIS HASSELBLATT, AND ANNE VAUGON

is presented in section 3. The Reeb flow RαA is Anosov if q is positive—and only then (theorem 3.3). Definition 2.1 ([37]). Let M be a manifold and ϕ: R M M a smooth flow with nowhere vanishing generating vector field X. Then× ϕ (and→ also X) is said to be an Anosov flow if the tangent bundle TM (necessarily invariantly) splits as TM = RX E+ E− (the flow, strong-unstable and strong-stable directions, respectively), in⊕ such⊕ a way that there are constants C > 0 and η > 1 > λ > 0 for which −t + −t t − t (1) Dϕ E Cη and Dϕ E Cλ ≤ ≤ for t > 0. The weak-unstable and weak-stable bundles are RX E+ and RX E−, respectively. (E± are then tangent to continuous foliations ⊕W ± with smooth⊕ leaves.) An Anosov flow on a 3-manifold is said to be of algebraic type if it is finitely covered by the geodesic flow of a surface of constant negative curvature or the suspension of a diffeomorphism of the 2-torus, and it is called a contact Anosov flow if it is a Reeb flow, in which case E+ E− is the contact structure and α is said to be Anosov as well. Geodesic flows of⊕ Riemannian manifolds with negative sectional curvature are Anosov flows. For surfaces of constant negative curvature it is easy to verify the defining property directly, and we do so at the start of section 2.4.3. In this paper, we show that the complexity of the resulting flow exceeds that of the flow on which the surgery is performed. We measure the complexity of the flow of X via its orbit growth, entropy and cohomological pressure. For a contact ρ form α, a free homotopy class ρ and T > 0, we denote by NT (α) the number of Rα-periodic orbits in ρ with period smaller than T and NT (α) the number of Rα-periodic orbits with period smaller than T . The orbit growth of Rα (or the associated flow) is the asymptotic behavior of NT (α), its exponential growth rate is the topological entropy. We summarize the needed notions and facts in section 2.3.1. Cohomological pressure drives orbit growth in a given homology class and is defined in section 2.3.2. 2.2. New contact flows. We begin with a paraphrase of the main result of the surgery construction from [29] in a way that points to the broader perspective of the present work and make a few initial observations that go further. Theorem 2.2 ([29, Theorems 1.6, 1.9]). On the unit tangent bundle M of a negatively curved surface, there is a family of smooth Dehn surgeries, including the Handel–Thurston surgery, that produce new contact flows. The surgered geodesic flow has the following properties: (1) It acts on a manifold that is not a unit tangent bundle. (2) If it is Anosov, it is not ortibt equivalent to an algebraic Anosov flow. (3) If it is Anosov, then its topological and volume entropies differ, or, equiva- lently, the measure of maximal entropy is always singular [28]. (4) If it is Anosov and if the surgered manifold is hyperbolic, then non empty free homotopy class ρ of closed orbits is infinite and is an isotopy class,2 moreover, there exist a1, c1, a2, c2 > 0 such that

1 ρ ln(T ) c2 NT (αA) a1 ln(T ) + c1 a2 − ≤ ≤

2Each closed orbit is related to at most ﬁnitely many others by the pair being the boundary of an embedded cylinder [9]. (This relation is neither transitive nor reﬂexive.) For comparison, isotopy only ensures that the circles in question are the boundary components of an immersed cylinder. ORBIT GROWTH OF CONTACT STRUCTURES AFTER SURGERY 5

for all T > 0, where αA is the contact form defined on the surgered manifold. [26, Theorem A], [8, Remark 5.1.16, Theorem 5.3.3], [9], [10, Theo- rem F]. That these surgeries produce contact flows on hyperbolic manifolds is a corollary of the two following theorems. Theorem 2.3 (Thurston [51, 52, Theorem 5.8.2], Petronio and Porti [47]). For all but infinitely many slopes, Dehn filling a hyperbolic 3-manifold gives rise to a hyperbolic manifold. Theorem 2.4 (Folklore [29, Theorem 1.12]). Suppose Σ is a hyperbolic surface, π : SΣ Σ its unit tangent bundle, γ : S1 M continuous such that c := π γ is a closed→ geodesic that is not the same geodesic→ traversed more than once and◦ such that ` c = ∅ whenever ` is a noncontractible closed curve. Then SΣ r (γ(S1)) is a hyperbolic∩ 6 manifold. Nonetheless, there exist infinitely many closed orientable hyperbolic manifolds of dimension 3 which do not support an Anosov flow [49, Theorem A]. Additionally, since there are only finitely many homotopy classes of tight contact structures on a 3-manifold [17, Théorème1] and the contact structures with an Anosov Reeb flow are tight as they are hypertight ([48], [6, p. 18]), there exist only finitely many homotopy classes of contact Anosov flows on a given 3-manifold. On hyperbolic 3-manifolds the same goes for isotopy classes [17, Théorème2]. We do not know if the surgery from theorem 2.2 can produce different contact structures on the same manifold. Remark 2.5. The dynamical properties of the flow after surgery differ from the properties of Anosov algebraic flows. Indeed, for algebraic flows, free homotopy classes of closed orbits are finite. For geodesic flows no two (parametrized) orbits are homotopic, though rotating the tangent vector through π isotopes each to its flip, which has the same image as another orbit (the same geodesic run backwards), and only in suspensions are all free homotopy classes of images of orbits singletons [10, Corollary 4.3]. Our surgery corresponds to a (1, q)-Dehn surgery and produces Anosov Reeb flows for q > 0. As part of our study− focuses on the q < 0-case, it is important to note the following. Proposition 2.6. Some surgeries from theorem 2.2 produce flows that are not Anosov (theorem 3.3). In the case q = 1, this surgery is the standard Weinstein surgery as defined by Weinstein [55] in 1991 simplifying Eliashberg’s work [22] of 1990 (see [30, Chapter 6] for more details). The surgery (1, q) for any q can be deduced from this construction. A direct construction for any q using Giroux theory of convex surfaces can be found in [19]. In answer to a question of Serge Troubetzkoy we here note: Proposition 2.7. There are analytic Anosov flows as described in theorem 2.2. Proof. The contact form is smooth and can hence be approximated by analytic ones. The contact property of the form and the Anosov property of its Reeb flow are open. Remark 2.8. Another perspective on the connection with the Handel–Thurston construction is that our result implies in particular that the Handel–Thurston examples are topologically orbit-equivalent to contact flows. 6 PATRICK FOULON, BORIS HASSELBLATT, AND ANNE VAUGON

Remark 2.9. For context we recall here that contact Anosov flows have the Bernoulli property [36, 45, 16] and exponential decay of correlations [39]. The Bernoulli property and the Ornstein Isomorphism Theorem [44] imply that the flows we obtain from our surgery are measure-theoretically isomorphic to the original contact Anosov flow up to a constant rescaling of time, the constant being the ratio of the Liouville entropies. (This answers a question of Vershik.) 2.3. Production of closed orbits for contact Anosov flows.

2.3.1. Impact on entropy. We continue with new results about the features of the contact Anosov flows from [29] to the effect that the surgery of theorem 2.2 produces “exponentially many” closed orbits. We preface these statements by a brief summary of the needed notions and facts pertinent to entropy. The topological entropy of an Anosov flow (or of the vector field that gener- • ates it) equals the exponential growth rate of the number of periodic orbits; 1 in our case this means that htop(Rα) = limT →∞ T log NT (α). t t The entropy hµ(ϕ ) of a flow ϕ with respect to an invariant Borel prob- • ability measure µ (also referred to as the entropy of µ with respect to ϕt) does not exceed the topological entropy of ϕt.3 If a flow-invariant Borel probability measure µ is absolutely continuous with • respect to a smooth volume, then we say it is a Liouville measure and write

hLiouville := hµ. t t t For the geodesic ﬂow g of a surface we have hLiouville(g ) = htop(g ) if (and • only if [28, 34, 35]) the curvature is constant. Scaling of time: if s (0, ), then h (sX) = sh (X) and • ∈ ∞ Liouville Liouville htop(sX) = shtop(X). More generally, there is Abramov’s formula: the entropy of a time change • gX of a nonzero vector ﬁeld X with respect to a gX-invariant probability measure µg canonically associated with an X-invariant Borel probability measure µ is Z

(2) hµg (gX) = hµ(X) g dµ.

This means that comparisons of the intrinsic dynamical complexity of these vector fields are meaningful only when R g = 1. Pesin entropy formula [7]: For a volume-preserving flow ϕt with 1-dimen- • t sional expanding direction, hLiouville(ϕ ) equals the positive Lyapunov exponent of the flow [7], [37, Definition S.2.5], which is (a.e.) defined as the exponential growth rate of unstable vectors under the flow and as a function of time. Theorem 2.10. If ψt is a contact Anosov flow obtained from the geodesic flow gt of a compact oriented surface of constant negative curvature by the surgery in theorem 2.2 (generated by the vector field in (8)), then its topological entropy is strictly larger. Indeed, h (ψt) > h (ψt) h (gt) = h (gt). top Liouville ≥ Liouville top Since htop measures the exponential growth rate of periodic orbits of a hyperbolic t t dynamical system, the number NT (ψ ) of ψ -periodic orbits of period t T (of up t ≤ to a given length) grows at a larger exponential rate than NT (g ). Remark 2.11. The strict inequality in theorem 2.10 is obtained by contraposition of a rigidity result [28], so we do not know by how much the topological entropy

3Indeed, the topological entropy is the supremum of the entropies of invariant Borel probability measures (Variational Principle). ORBIT GROWTH OF CONTACT STRUCTURES AFTER SURGERY 7 increases through our surgery. Recently, Bishop, Hughes, Vinhage and Yang suggested to provide effective lower bounds for this entropy-increase by using cutting sequences in the spirit of Series. 2.3.2. Growth in homology classes. In a self-contained digression, we can give rather more detailed information about orbit growth in homology classes. Theorem 2.12. If ψt is a contact Anosov flow obtained from the geodesic flow gt of a compact oriented surface of constant negative curvature by the surgery in theorem 2.2 (generated by the vector field in (8)), then ζ t η t exponentially NT (ψ ) NT (g ) −−−−−−−→T →∞ ∞ t t ζ t η t for any homology classes ζ for ψ and η for g (where NT (ψ ) and NT (g ) count the number of periodic orbits orbit with period T in the homology classes ζ and η). ≤ The proof derives from the notion of cohomological pressure. Definition 2.13 ([50, Theorem 1(iii), p. 398]). The cohomological pressure of ϕt is Z t n t o P (ϕ ) := inf sup hµ(ϕ ) + b(X) dµ , 1 [b]∈H (M,R) µ∈M(ϕt) where (ϕt) is the set of ϕt-invariant Borel probability measures. M Remark 2.14. The cohomological pressure is the usual pressure of the function b(X) and the abose formula is well-defined for a cohomology class [b]. Indeed, here, H1(M, R) is the first de Rham cohomology group, and the integral is the Schwartzman winding cycle, which is well-defined for a closed 1-form when µ is ϕt-invariant; the supremum is unaffected by addition of an exact form to b. Contact Anosov flows satisfy (3) h (ϕt) P (ϕt) h (ϕt) [25, Corollary 1]. top ≥ ≥ Liouville Theorem 2.15 ([25, Theorem 5.3]). If ϕt is a flow such as the ones obtained in t t theorem 2.2, then P (ϕ ) > hLiouville(ϕ ). Thus, the conclusion of theorem 2.10 is strengthened to h (ϕt) P (ϕt) > h (ϕt) h (gt) = h (gt). top ≥ Liouville ≥ Liouville top This makes it possible to amplify the observation about increased orbit growth and prove theorem 2.12. Indeed, contact Anosov flows are homologically full4 [25, Proposition 1], and, for homologically full flows, cohomological pressure drives orbit growth in a given homology class ζ [50, Theorem 1]:

TP (ϕt) ζ e (4) N (ϕt) C(ζ) as T , T 1+ b1 ∼ T 2 → ∞ where b1 is the first Betti number of the underlying manifold. 2.4. Production of closed orbits for any Reeb flow. We now broaden the scope far beyond hyperbolic dynamics by beginning to involve contact geometry in a serious fashion. Specifically, the existence of well-understood Reeb flows, such as those in theorem 2.2, allows us to control all the other Reeb flows associated to the same contact structure in terms of entropy or orbit growth. We transcend hyperbolicity because we describe here our results concerning dynamical properties of Reeb flows associated to all (or a subclass of) contact forms after a contact surgery. These flows need not be hyperbolic even if the contact structure arises from an Anosov flow.

4I.e., every homology class contains a closed orbit 8 PATRICK FOULON, BORIS HASSELBLATT, AND ANNE VAUGON

2.4.1. Orbit growth from Anosov Reeb flows. This section presents an archetype of theorem deriving properties for all Reeb flows from stronger properties for one Reeb flow. Our results described in section 2.4.2 can be seen as extension of this theorem. It can be applied to some of the contact flows described in theorem 2.2. The existence of Anosov Reeb flows is a source of exponential orbit growth for all Reeb flows as proved by Alves or Macarini and Paternain [40, Theorem 2.12.]. Theorem 2.16 (Alves [3, Corollary 1]). If one Reeb flow for a compact contact 3-manifold (M, ξ) is Anosov, then every Reeb flow on (M, ξ) has positive topological entropy.

Remark 2.17. Alves also obtains lower bounds for the entropy: for α = fα0 with f > 0, we get h(Rα) a/ max(f) where a is some growth rate associated to Rα0 . Note also that these estimates≥ can not be obtained by the Abramov formula, which determines the measure-theoretic entropy of a time-change because different Reeb fields for a contact structure need not be collinear. The standard contact structure on the unit tangent bundle of a hyperbolic surface has an Anosov Reeb flow and therefore, by theorem 2.16, all its other Reeb flows have positive entropy and their orbit growth is at least exponential. In particular, theorem 2.16 applies to the contact structures obtained in theorem 2.2 on hyperbolic manifolds: these are examples satisfying the Colin–Honda conjecture, and on nonhyperbolic manifolds, for instance, when the surgery is associated to a simple geodesic. We give a slightly different proof of this result in section 4. 2.4.2. Orbit growth from contact homology. We now present our results and extend theorem 2.16 in two different settings (1) when the Reeb flow after surgery is Anosov, we study orbit growth in free homotopy classes; (2) when the geodesic associated to the surgery is a simple curve, we prove positivity of entropy for any contact form (and any surgery). Let us describe our results in the first setting. The following result can be seen as a corollary of the invariance of contact homology and the Barthelmé–Fenley estimates from [10, Theorem F] in the nondegenerate case, and of Alves’ proof of Theorem 1 in [3] and the Barthelmé–Fenley estimates from [10, Theorem F] in the degenerate case.

Theorem 2.18. Let (MS, ξS = ker(αA)) be a contact manifold obtained after a non-trivial contact surgery such that αA is Anosov. Let ρ be a primitive free homotopy class containing at least one RαA -periodic orbit. Then for all contact forms α on (MS, ξS), ρ contains infinitely many Rα-periodic orbits. Additionally, ρ (1) if α is nondegenerate, there exist a > 0 and b R such that NT (λ) a ln(T ) + b for all T > 0, ∈ ≥ (2) if α is degenerate and MS is hyperbolic, there exist a > 0 and b R such that N ρ (λ) a ln(ln(T )) + b for all T > 0. ∈ T ≥ ρ ρ Remark 2.19. In fact, for α nondegenerate, we will prove NT (α) NCT (αA) for some C > 0 and for all T > 0 and use the Barthelmé–Fenley result. Therefore≥ better ρ control of NT (αA) in some free homotopy classes will lead to better estimates. ρ Remark 2.20. There is no hope to obtain a upper bound on NT (αA) for all contact forms as the number of Reeb periodic orbits can always be increased by creating many periodic orbits in a neighborhood of a preexisting periodic orbit. Remark 2.21. If the manifold is not hyperbolic, the Barthelméand Fenley estimates are weaker as the upper bound is linear. The proof of theorem 2.18 can be ORBIT GROWTH OF CONTACT STRUCTURES AFTER SURGERY 9 adapted to this situation but leads to weak control of the growth of periodic orbits in a given homotopy class for degenerate contact forms. We now turn to our second setting and assume the geodesic associated to the surgery is a simple curve. Note that we do not assume that the Reeb flow is Anosov and therefore consider any (1, q)-Dehn surgery. Additionally, note that MS is never a hyperbolic manifold in this setting. Our main theorem is the following.

Theorem 2.22. If (MS, αA) is a contact manifold obtained from contact surgery along a simple geodesic, then any Reeb flow of (MS, ker(αA)) has positive topological entropy. In particular, the number of periodic orbits grows at least exponentially with respect to the period. The proof of this theorem is based on Alves’ work [1]. In the same paper, Alves obtains the same result when the associated geodesic is separating [1, Section 4 and Theorem 2]. Our strategy of proof is similar to that of Alves. Floer type homology and especially contact homology are the main tools to control Reeb periodic orbits of all contact forms associated to a contact structure. The contact homology of a “nice” contact form α0 is the homology of a complex generated by Rα0 -periodic orbits and therefore encode dynamical properties of the Reeb vector field (contact homology is described in section 4). The growth rate of contact homology makes it possible define the polynomial behavior of a contact structure. We now focus on examples obtained by surgery exhibiting polynomial growth.

2.4.3. Coexistence of diverse contact flows. We first introduce the three Reeb flows that naturally appear on the unit tangent bundle of a constantly curved surface of higher genus. This is elementary but not commonly presented. On the unit tangent bundle of a hyperbolic surface, there is a canonical framing consisting of X, the vector field on SΣ that generates the geodesic flow, of V , the vertical vector field (pointing in the fiber direction), and of H := [V,X]. It satisfies the classical structure equations (5) [V,X] = H, [H,X] = V, [H,V ] = X.

One can check these by using that in the PSL(2, R)-representation of SΣ,e these vector fields are given by 1/2 0 0 1/2 0 1/2 X ,H ,V − . ∼ 0 1/2 ∼ 1/2 0 ∼ 1/2 0 − The structure equations imply that e± := V H satisfies [X,V H] = e±, so if a vector field f e± along an orbit of X is± invariant under the± geodesic∓ flow, then 0 = [X, fe±]· = (f˙ f)e±, where f˙ is the derivative along the orbit. This ∓ means that f˙ = f, so f(t) = const e±t. Thus, the differential of the geodesic flow expands and contracts,± respectively, the directions e±; this is the Anosov property and E± is spanned by the vector e± = V H. ± Of course, in the PSL(2, R)-representation of SΣ,e these 3 flows are given by

t/2 1/2 0 e 0 X exp t = t , 0 1/2 0 e− /2 − 0 1/2 cosh t/2 sinh t/2 H exp t = , 1/2 0 sinh t/2 cosh t/2 0 1/2 cos t/2 sin t/2 V exp − t = − 1/2 0 sin t/2 cos t/2 10 PATRICK FOULON, BORIS HASSELBLATT, AND ANNE VAUGON

To see in these terms that X generates a contact ﬂow, deﬁne a 1-form α0 by α0(X) = 1 and α0(V ) = 0 = α0(H). For Z V,H we have ∈ { } ≡0 ≡1 ∈−{V,H}

d α0(X,Z) = X α0(Z) Z α0(X) α0([X,Z]) = 0, L − L − =0 =0 =0 so ιX d α0 0. Additionally α0 d α0(X,V,H) = α0(X) d α0(V,H) = 1 because ≡ ∧ ≡0 ≡0 =−X

d α0(V,H) = V α0(H) H α0(V ) α0([V,H]) = 1. L − L − =0 =0 =−1

Thus, α0 d α0 is a volume form; in fact a volume particularly well adapted to this ∧ canonical framing, and α0 is a contact form. Additionally, X = Rα0 . Likewise, one can check that the 1-forms β and γ defined by β(V ) = 1 and β(X) = 0 = β(H), and γ(H) = 1 and γ(X) = 0 = γ(V ) are also contact forms. Their Reeb vector fields are Rβ = V and Rγ = H. Note that γ = dα0(V, ) and · β = dα0(H, ). Additionally, the orientation given by β dβ is the opposite − · ∧ of the orientation given by α0 d α0; therefore α0 and β define different contact ∧ t structures. By contrast, α0 and γ define isotopic contact structures. Indeed, let ψ be the flow of V . Then,

t t (ψt)∗X = cos /2X + sin /2H and t t (ψt)∗H = cos /2H sin /2X, − thus t t (ψt)∗α0 = cos /2α0 + sin /2γ as the two contact forms coincide on (ψt)∗X,(ψt)∗H and (ψt)∗V = V . So it suffices to study the geodesic flow as the leading representative of this S1-family of contact Anosov flows. Geometrically, this family of flows can be described as: rotate a vector by an angle, carry it along the geodesic it now defines, and rotate back by the same angle. In other words, it is parallel transport for a fixed angle.

Dynamically Rα0 and Rβ are polar opposites: the geodesic flow is hyperbolic and the fiber flow is periodic. The surgery increases the complexity of both, whether or not the twist goes in the correct direction to produce hyperbolicity from the geodesic flow. For the geodesic flow this is theorem 2.22, and for the fiber flow it is:

Theorem 2.23. Let (MS, ker(βS)) be a contact manifold obtained from the contact form for the fiber flow after a non-trivial contact surgery along a simple geodesic. Then the growth rate of contact homology for (MS, ker(βS)) is quadratic. In particular, any nondegenerate Reeb flow of (MS, ker(βS)) has at least quadratic orbit growth5. 2.4.4. Relation to other works on contact surgery and Reeb dynamics. Weinstein surgery/handle attachment is an elementary building block and fundamental operation in contact/symplectic topology and has been largely studied from the topological point of view (for instance it can be used to construct specific or tight or fillable contact manifolds). We only mention here works focusing on the Reeb dynamics. A description of contact surgery with control of the Reeb vector field can be found in [24], where Etnyre and Ghrist construct tight contact structures and prove tightness using dynamical properties of the Reeb vector field (their desciption is

5 This means that for any nondegenerate contact form β such that ker(β) = ker(βS ) the number 2 NT (β) of Rβ -periodic orbits with period smaller than T satisfies NT (β) ≥ aT for some positive real number a. ORBIT GROWTH OF CONTACT STRUCTURES AFTER SURGERY 11 different from ours as they consider a surgery on a transverse knot and focus on the description of this surgery via tori). In [14], Bourgeois, Ekholm and Eliashberg describe the effect of a Weinstein surgery on Reeb dynamics and contact homology. More precisely, they prove the existence of an exact triangle in any dimension connecting contact homologies of the initial manifold and the surgered manifold and a third term associated to the attaching sphere and called Legendrian contact homology. However, explicit computations are delicate even for our explicit examples, for instance as the Lengendrian contact homology is the homology of a huge complex. In contrast, our results give precise estimates in Reeb dynamics but for specific examples. Our work is largely inspired by Alves’ work on Reeb dynamics as explained above, note that he himself applied his methods to contact sugery. The study of Reeb flows with positive entropy comes from Macarini and Schlenk [41] of the unit cotangent bundle equipped with the standard contact structure. This has been developed by Macarini and Paternain [40], Alves [1, 2, 3] and others. In [4], Alves, Colin and Honda relate topological entropy of Reeb flows to the monodromy of an associated open book decomposition.

3. Surgery and production of closed orbits The surgery in [29] on which this work is based came with some infelicitous conventions and an immaterial sign error, so we recapitulate some of the steps here with more explicit details. This is necessary also as a base for the proof of theorem 2.10, and for a supplementary result (theorem 3.4) that is needed later. Our surgery can be performed in a neighborhood of any Legendrian knot in a contact 3-manifold. We start with a description of the surgery in adapted coordinates near a Legendrian and then explain how to obtain such coordinates in the unit tangent bundle of a hyperbolic surface and how they are linked to the stable and unstable bundles.

3.1. The surgery from the contact viewpoint. Let (M, ξ = ker(α)) be a contact 3-manifold and let γ be a Legendrian knot in M. Then there exist coordinates (t, s, w) Ω := ( η, η) S1 ( , +), ∈ − × × − with 0 < < η/2π on a neighborhood of γ in which α = dt + w ds and γ = 0 S1 0 . The surgery annulus is 0 S1 ( , +). Note that in these { } × × { } { } × ∂ × − coordinates α dα = t. dw ds and Rα = ∂t , so Ω is a flow-box chart. The surgeries split∧ this chart∧ into∧ 2 one-sided flow-box neighborhoods of the surgery annulus, and while the initial transition map between these on 0 S1 ( , +) { } × × − is the identity, the surgered manifold MS is defined by imposing the desired twist (or shear) as the transition map on this annulus: (6) F : S1 ( , ) S1 ( , ), (s, w) (s + f(w), w) × − → × − 7→ with f :[ , ] S1, w exp(iqg(w/)), q Z, g : R [0, 2π] nondecreasing smooth, 0− g0 → 4 even,7→ and g(( , 1]) = ∈0 , g([1, →)) = 2π . We specify that the transition≤ ≤ map from t <−∞0 to− t > 0{ }is used∞ to identify{ } points (0−, x) with (0+,F (x)). With this choice{ one} see{ that}F ∗α = α + wf 0(w) dw and hence that F ∗ dα = dα and F ∗(α dα) = α dα, ∧ ∧ so α dα is a well-defined volume on MS. The vector field Rα on M induces the ∧ Handel–Thurston vector field XHT on MS. Its flow preserves the Liouville volume defined by α dα [29, Corollary 3.3], and the total volume of the manifold is not changed by the∧ surgery. 12 PATRICK FOULON, BORIS HASSELBLATT, AND ANNE VAUGON

However, we have not yet produced a contact ﬂow: F ∗α = α + wf 0(w) dw, so α does not induce a contact form on MS. A deformation yields a well-deﬁned contact form α∓ = α dh for t 0, where h ∓ ± ≥ 1 Z w h(t, w) := λ(t) xf 0(x) dx on ( η, η) ( , ) and h = 0 outside. 2 − − × − λ: R→[0,1] is a smooth bump function

1 0 satisﬁes dh = 2 wf (w) dw on the surgery annulus and h 0 for t close to η. ∗ + − ± ≡ ± Hence F (αh ) = αh and αh induces a contact form αA on MS. Its Reeb ﬁeld is a time-change

XHT (7) RαA := 1 dh(XHT ) ± of XHT [29, Theorem 4.2], which is well-defined because dh(XHT ) < 1 if 0 < < η/2π [29, Theorem 4.1]. If one considers smaller , it| is possible| to impose the condition dh(XHT ) < 1/2 and we will do so in section 7. | | The time-change that defines RαA is a slow-down near the surgery annulus, which confounds comparisons of dynamical complexity because of the extra factor in Abramov’s formula (2), so we study the vector field

(8) Xh := cRαA = RαA/c, Z c where c R is such that α dα = 1 to compare entropies. ∈ 1 dh(XHT ) ∧ ± 3.2. Surgery on unit tangent bundle and Anosov flows. We now explain how to perform a contact surgery on the unit tangent bundle of a hyperbolic surface Σ. Select a closed geodesic c: S1 Σ, s c(s) and consider the Legendrian knot γ obtained by rotating the unit vector→ field7→ along c by the angle θ = π/2. This knot is Legendrain Hasselblatt as H Boris is and tangent Foulon to γ (seePatrick fig. 1). Standard coordinates for αA near γ are1240

Figure vectors normal 1. Surgery and annulus geodesic closed in the simple A base 5: Figure obtained tangent by the flowingwith ✓ along angle the the verticaland c field geodesic V the andof s then along parameter the the geodesic by vectorparametrized fieldc to X [29, Lemma perpendicular 5.1]: field the vector surgery unit the annulus is knot is contained Legendrian in The the torus geodesic. T the above of cvector (see fig. 2); it consists of vectors that are almost orthogonal to a. 2 chosen= ⇡ geodesic✓ by ingiven a surface. Along γ, E+ is spanned by a vector V + H in the first quadrant. D To prove that the surgered flow is Anosov, [29] uses Lyapunov–Lorentz metrics [29, Claim 4.5 and Appendix A].

eoesurgery before 1 S 1 S ✏ ⇡ ; ✏ ⇡ 1 S annulus The 6: Figure ⇥ ⇢ C 2 2 ⇥ nietetorus, the inside 2 = ⇡ ✓ around annulus an in surgery the localize we Although D . ] 28 [ in as same the clearly is this topologically linearize , c of ﬁeld vector unit perpendicular the of ƒ neighborhood a parametrize To , 2 = ⇡ near ✓ for ✓ cos ` w taking by c to ﬁeld vector tangent the with ✓ angle the ⇡ 2 WD hsgvsparameters gives This . c of length the is ` where /; ✏ ; ✏ . 1 S / ⌘ ; ⌘ . ;w/ s ; t . (4) C ⇥ ⇥ WD 2

oue1 (2013) 17 Volume , opology & eometry T G Surgery in phase space Surgery in phase space

Contact ORBIT GROWTHContact OF CONTACT STRUCTURES AFTER SURGERY 13 Anosov ﬂows Anosov ﬂows on hyperbolic on hyperbolic 3-manifolds 3-manifolds Patrick Foulon, Patrick Foulon, Boris Boris Hasselblatt, Hasselblatt, Anne Vaugon Anne Vaugon

Introduction Introduction

Summary of results Summary of results

Deﬁnitions Deﬁnitions

Statement of results Statement of results Comments Comments

Proofs Proofs

Surgery Surgery

Local computations Local computations

Intrinsic Intrinsic complexity of complexity of the contact the contact structure structure Hyperbolic 1 Hyperbolic 1 1 1 1 1 Annulus Σ S ( π/2 ϵ, π/2manifoldsϵ) S S beforeAnnulus surgeryΣ S ( π/2 ϵ, π/2 ϵ) S S after surgery manifolds = × −Figure− − 2. +Surgery⊂ × annulus before= and× after− − surgery− + (q =⊂ 1)×

Definition 3.1. The continuous Lorentz metrics Q+ and Q− on M are a pair of Lyapunov–Lorentz metrics for the flow ϕt generated by X if there exists constants a, b, c, T > 0 such that (1) C+ C− = ∅ where C± is the Q±-positive cone; (2) Q±(∩X) = c; − ± ± ±t bt ± (3) for any x M, v C (x) and t > T , Q (Dxϕ (v)) ae Q (v) ∈ ∈ ≥ ±T ± ± ±T (4) for any x MDxϕ C (x) r 0 C ϕ (x) ∈ { } ⊂ Proposition 3.2. [29, Claim 4.5 and Appendix A] A smooth flow ϕt is Anosov if and only if it admits a pair of Lyapunov–Lorentz metrics Q− and Q+. The unstable foliation of the flow is then contained in the positive cone Q+ and the stable foliation in the positive cone of Q− For the geodesic flow, one can choose Q± = dw ds c dt2 in the coordinates (t, s, w). Understanding how the surgery affects± the positive− cones of Q± is crucial to understand why the condition q positive is essential to obtain an Anosov flow after surgery. We restrict attention to the trace of these cones in the sw-plane and consider the geometry of the action of F by differentiating (6) to see the twist (shear) in (s, w)-coordinates: 1 f 0(w) DF = . 0 1 Therefore, if q > 0, the image of the first and third quadrant (ie the trace of C+)

e+ e+ + C+ DF (C )

+ DF (C+) C

Figure 3. Action of a positive and negative twist (shear) on the ﬁrst quadrant

is a subcone of the ﬁrst and third quadrant that shares the horizontal axis (see 14 PATRICK FOULON, BORIS HASSELBLATT, AND ANNE VAUGON

fig. 3). Roughly speaking, this implies that the cone field C+ is preserved by the surgery and one can define a new cone field one the surgered manifold by Q± = dw ds c dt2, if t < 0, 0 ± − Q± = dw ds b(t)f 0(w) dw2 c dt2, if t > 0, 1 ± − − where b: R R+ smooth with b(( , 0]) = 1 , b([η, )) = 0 and b0 < 0 on → ∗ −∞ ± { } ± ∞ { } ( η, η). Then for t = 0, F Q1 = Q0 and Q and Q induces a pair of Lyapunov– − 0 1 Lorentz metrics on MS. If q < 0, the cones are not preserved and the flow is not Anosov: Proposition 3.3. The (1, q)-Dehn surgery defined by F in (6) does not produce an Anosov flow if q/ is large enough, i.e., if either q < 0 is fixed and is small enough or if > 0 −is fixed and q < 0 with q big enough. | | Proof. There is a lower bound on the return time to the surgery region, so there is a K > 0 such that the half-cone a Kb 0 is mapped into the half-cone 0 a Kb by the differential of the≤ return− ≤ map (see fig. 4). Here, we use coordinates≥ ≥ (a, b) in the (s, w)-plane. Now suppose that q/ < 2K and that the function g in the definition of f (after (6)) is chosen with monotone− derivative on (0, ). Then f 0(0) < q/, so f 0(w) < q/ for small w. This has the effect that for∞ such w, the half-cone around e+ given by 0 a Kb is mapped by DF into the half-cone a Kb 0, which is on the other≤ ≤ side of e−. The return map then sends it into≤ the − half-cone≤ 0 a Kb, which is the other half of the cone in which we started. This is incompatible≥ ≥ with the existence of a continuous invariant cone field that extends to points that miss the surgery region, and hence with the Anosov property.

e+ e+ e+ 0 ≤ a ≤ Kb

return shear map a = Kb −−−→ a −−−−→ ≤−Kb ≤0 a = a = Kb −Kb

0 ≥ a ≥ Kb e− e− e−

Figure 4. The cones in the proof of theorem 3.3

Note that one can perform a positive surgery on an Anosov flow (and therefore obtain another Anosov flow) then undo it by performing a negative surgery and obtain again an Anosov flow. This is compatible with the satement of theorem 3.3, as q and are fixed in the negative surgery (and thus theorem 3.3 does not apply). Returning to the case of positive q, we note from the preceding:

Proposition 3.4. The stable and unstable foliations of (MS, αA) as described in theorem 2.2 are orientable. Proof. The strong stable foliation is contained in the positive cone of Q− and the strong unstable foliation in the positive cone of Q+, so the stable foliation is orientable if and only if the positive cone of Q− is orientable (an orientation of the positive cone is a choice of a connected component of this cone). The stable and unstable foliations of the unit tangent bundle over a hyperbolic surface are − ∂ ∂ ∗ ∂ ∂ orientable. Additionally, Q ∂s , ∂s = 0 and F ∂s = ∂s , so the surgery preserves − − + the orientation of Q , and Q is orientable. It implies that Q is orientable. ORBIT GROWTH OF CONTACT STRUCTURES AFTER SURGERY 15

3.3. Impact on entropy. The nature of the surgery map then implies:

Proposition 3.5. If q 0, then hLiouville(XHT ) hLiouville(X). ≥ ≥ Proof. By the Pesin entropy formula it suffices to show that the positive Lyapunov exponent of XHT is no less than that of X. Volume-preserving Anosov 3-flows are ergodic [37, Theorem 20.4.1], so the positive Lyapunov exponent, being a flow- invariant bounded measurable function, is a.e. a constant. The earlier observation that for the geodesic flow on a hyperbolic surface the expanding vector is of the form ete+ means that the Lyapunov exponent of (the normalized) Liouville measure is 1. Therefore, we will show that the positive Lyapunov exponent of XHT is at least 1. To that end we verify that the differential of its time-1 map expands unstable vectors by at least a factor of e with respect to a suitable norm. For the geodesic flow the Sasaki metric induces a natural norm, and this norm is what is called an adapted or Lyapunov norm: for unstable vectors, this norm grows by exactly et under the flow, and on each tangent space it is a product norm. Our argument involves only vectors in unstable cones, so we pass to a norm + that is (uniformly) equivalent when restricted to such vectors: the norm of thek · unstable k component. Geometrically, this means that at each point we project tangent vectors to E+ along E− RX and take the length of this unstable projection as the norm ⊕ t t of the vector. Thus, Dg (v) + = e v + for t 0. k k k k ≥ The proof of hyperbolicity of XHT shows that the cone field defined by the Lyapunov–Lorentz functions is well-defined on the surgered manifold and invariant under XHT . Thus, this adapted norm for the geodesic flow defines a (bounded, t though discontinuous) norm + on unstable vectors for the flow ϕ defined byXHT . 1 k·k We now show that Dϕ (v) + e v + for any v in an unstable cone. This is clear (with equality)k when thek underlying≥ k k orbit segment does not meet the surgery annulus because the action is that of the geodesic flow. If there is an encounter with 0 t 0 t the surgery annulus at time t (0, 1], then v := Dϕ (v) satisfies v + = e v +, 00 ∈ 0 00 0 k k k k and we will check that v := DF (v ) satisfies v + v , which implies that 1 1−t 00 1−t 00 1k−t k0 ≥ k 1k−t t Dϕ (v) + = Dϕ (v ) + = e v + e v + = e e v + = e v +, as required.k k k k k k ≥ k k k k k k 0 0 That DF (v ) + v follows from the same argument as hyperbolicity of k k ≥ k k XHT as suggested by fig. 5, which superimposes the tangent spaces at some x and F (x) in the surgery annulus (using the identification from the canonical isometries between these tangent spaces). DF is a positive shear, and in the H-V -frame in ∂ the figure the addition of a multiple of the projection of ∂s (which is close to H) by a positive shear results in an increase in the projection to E+, which is spanned by e+ = V + H.

-increment + e

e − H = + V V − v0 = 0 H + ) e (v 00 = DF v H

Figure 5. DF increases the unstable component 16 PATRICK FOULON, BORIS HASSELBLATT, AND ANNE VAUGON

Remark 3.6. Alternatively, let z be a point on the annulus of surgery such that its orbit under the flow Xs will cross the surgery region infinitely often. As usual consider z as the identification of p = (t, s, w) with F (p) = (t, s + f(w), w) if + − ξ0(F (p)) = ae (F (p)) + be (F (p)) is in the preserved cone given by a > 0, 0 b ≤ ≤ a. Consider the first return, at time t, Then if q = ϕt(F (p)) = (0, st, wt), the image of ξ0(F (p)) by the linear tangent map is 0 + 0 − 0 ξt(F (q)) = a e (F (q)) + b e (F (q)) + c X(F (q)) with 0 df a = a exp(t) + (wt)a0(a exp(t) b exp( t)), dw − − where ∂ (F (q)) = a e+(F (q)) + b e−(F (q)) + c X(F (q)) ∂s 0 0 0 0 df so a a(exp(t)+2 dw (wt)a0sht). This gives the desired inequality for the projected norm.≥

Remark 3.7. We emphasize that the entropy-increase is manifested for XHT and thus results from the surgery and not from the time-change that makes the flow contact. We are now ready to pursue the growth of periodic orbits. c Proof of theorem 2.10. Abramov’s formula (2) with g := and µg the 1 dh(XHT ) ± normalized volume defined by αA gives Z c hLiouville(Xh) = hLiouville(XHT ) α dα = hLiouville(XHT ). 1 dh(XHT ) ∧ ± Combined with our previous result, this gives t t (9) hLiouville(ϕ ) = hLiouville(Xh) = hLiouville(XHT ) hLiouville(X) = hLiouville(g ). | {z≥ } theorem 3.5 This in turn yields a comparison of topological entropies: (9) t z t }| t{ t htop(g ) = hLiouville(g ) hLiouville(ϕ ) < htop(ϕ ) . | {z } ≤ | {z } constant curvature theorem 2.2.3 Proof of theorem 2.12. By (4), increased cohomological pressure suffices: (3) (9) z t }| t t{ z t }| t{ t hLiouville(g ) P (g ) htop(g ) = hLiouville(g ) hLiouville(ϕ ) < P (ϕ ) . ≤ ≤ | {z } ≤ | {z } constant curvature theorem 2.15 Of course, applying (3) on the right-hand side reproves theorem 2.10. 4. Contact homology and its growth rate Contact homology is an invariant of the contact structure computed through a Reeb vector field and introduced in the vein of Morse and Floer homology by Eliashberg, Givental and Hofer in 2000 [23]. The definition of contact homology is subtle and complicated. In this paper, we will consider it as a black box and only use the properties of contact homology described in theorem 4.1.6

6Plus, for theorem 7.3 we also use (and detail in the proof) an elementary and standard application of the computation of contact homology in the Morse–Bott setting. ORBIT GROWTH OF CONTACT STRUCTURES AFTER SURGERY 17

Roughly speaking, contact homology is the homology of a complex generated by Reeb periodic orbits of a (nice) contact form. Yet the homology does not depend on the choice of a contact form (but it depends on the underlying contact structure). Therefore a Reeb vector field provides us with information on contact homology and vice-versa. The differential of this complex “counts” rigid holomorphic cylinders in the symplectization M R of our contact manifold (this is the technical part of the definition). These cylinders× are asymptotic to Reeb periodic orbits when the R-coordinate of the cylinder tends to . Roughly speaking, if a rigid cylinder ±∞ is asymptotic to γ± at , then it contributes 1 to the coefficient of γ− in the ±∞ ± differential of γ+. This can be seen as a generalization of the differential of Morse Homology where we “count” rigid gradient trajectories asymptotic to critical points of a Morse function. In particular, this implies that the differential of a periodic orbit only involves periodic orbits in the same free homotopy class and with smaller period. Moreover, the complex is graded and the differential decreases the degree by 1 (here we will only use the parity of this grading). Computing this differential is usually out of reach without a strong control of homotopic periodic Reeb orbits. Variants of contact homology can be defined by considering periodic orbits in specific free homotopy classes or periodic orbits with period bounded by a given positive real number T (this operation is called a filtration). In the later situation, the limit T recovers the original homology. This procees is fundamental to gather information→ ∞ on the growth rate of Reeb periodic orbits. We recall that, if γ is a nondegenerate T -periodic orbit of the Reeb flow ϕt of (M, ξ = ker(α)) and p is a point on γ, the orbit γ is said to be even if the T symplectomorphism dϕ (p):(ξp, dα) (ξp, dα) has two real positive eigenvalues, and odd otherwise. →

Theorem 4.1 (Fundamental properties of cylindrical contact homology). Let (M, ξ) be a closed hypertight contact 3-manifold, α0 a nondegenerate contact form on (M, ξ) and Λ a set of free homotopy classes of M, Λ (1) Cylindrical contact homology CHcyl(α0) is a Q-vector space. It can be of ﬁnite or inﬁnite dimension. It is the homology of a complex generated by

Rα0 -periodic orbits in Λ. (2) The differential of an odd (resp. even) orbit contains only even (resp. odd) orbits. Λ (3) If α is another nondegenerate contact form on (M, ξ), then CHcyl(α0) and Λ CHcyl(α) are isomorphic. Λ (4) There exists a filtered version CH≤T (α0) (for T 0) of contact homology: the associated complex is generated only by periodic≥ orbits in Λ with period Λ T . Therefore, CH≤T (α0) is a Q-vector space of finite dimension and ≤ Λ Λ dim CH≤T (α0) ] Rα0 -periodic orbits in Λ with period T =: NT (α0) ≤ { ≤ } Λ (5)( CH≤T (α))T is a directed system and its direct limit is the cylindrical contact homology. Having a directed system means that for all T T 0, there Λ Λ ≤ exists a morphism ϕT,T 0 : CH≤T (α0) CH≤T 0 (α0) and −→ ϕT,T = Id • if T0 T1 T2, then ϕT0,T2 = ϕT1,T2 ϕT0,T1 . • ≤Λ ≤ Λ ◦ As lim CH≤T (α0) = CH (α0), there exist morphisms

Λ Λ ϕT : CH≤T (α0) CH (α0) −→ such that the following diagram commutes for T T 0 ≤ 18 PATRICK FOULON, BORIS HASSELBLATT, AND ANNE VAUGON

ϕT,T 0 Λ Λ CH≤T (α0) CH≤T 0 (α0)

ϕT ϕT 0 Λ CH (α0)

(6) Let α = fα0 be another nondegenerate contact form. Assume f > 0, and let B be such that 1/B f(m) B for all m M. There exist C = C(B) and ≤Λ ≤ Λ ∈ morphisms ψT : CH≤T (α0) CH≤CT (α) such that the following diagram commutes −→ ψ Λ T Λ CH≤T (α0) CH≤CT 0 (α)

ϕT,T 0 (α0) ϕCT,CT 0 (α)

ψ 0 Λ T Λ CH≤T 0 (α0) CH≤CT (α)

This defines a morphism of directed system. Contact homology was introduced by Eliashberg, Givental and Hofer [23]. The filtration properties come from [18]. The description in terms of directed systems takes its inspiration from [43] and is presented in [54, Section 4]. Though commonly accepted, existence and invariance of contact homology remain unproven in general. This has been studied by many people using different techniques. This paper uses only proved results and follows Dragnev and Pardon approaches. If α is hypertight and Λ contains only primitive free homotopy classes, the properties of contact homology described in theorem 4.1 derive from [20] (see [54, Section 2.3]). In the general case, theorem 4.1 can be derived from [46]. Cylindrical contact homology for hypertight contact forms (and possibly nonprimitive homotopy classes) and the action filtration are described in [46, Section 1.8]. The case of a not hypertight contact form when there exists an hypertight contact form derives from the contact homology of contractible orbits [46, Section 1.8] and our invariant corresponds to Λ contr CH• . Note that when computed through a hypertight contact form, CH• is Λ trivial and CH• is the cylindrical contact homology. In the not hypertight case, our invariants can be interpreted geometrically using augmentations. This viewpoint is described in [54, Section 2.4 and Section 4]. Combining the two commutative diagrams from theorem 2.18 and the invariance of contact homology we obtain the following inequality.

Proposition 4.2. Let α0 and α = fα0 be two nondegenerate contact forms on (M, ξ), where M is a closed, 3-dimensional manifold and ξ is hypertight. Assume f > 0, and let B such that 1/B f(m) B for all m M. Then ≤ ≤ ∈ Λ N (α) rank(ϕL(α)) rank(ϕ (α0)) L ≥ ≥ L/C(B) for all L > 0. Λ If CH (α0) is well-understood, one can get an easier estimate.

Corollary 4.3. Let α0 and α = fα0 be two nondegenerate contact forms on (M, ξ) where M is a closed, 3-dimensional manifold and ξ is hypertight. Assume f > 0, and let B such that 1/B f(m) B for all m M. If ≤ ≤ ∈ Λ M CH (α0) = Qγ

Rα0 -Reeb periodic orbit γ in Λ Λ Λ then, N (α) N (α0) for all L > 0. L ≥ L/C(B) ORBIT GROWTH OF CONTACT STRUCTURES AFTER SURGERY 19

In fact, one can derive another invariant of contact structures from these properties of contact homology. Two nondecreasing functions f : R+ R+ and g : R+ → → R+ have the same growth rate type if there exists C > 0 such that x f g(x) f(Cx) C ≤ ≤ for all x R+ (for instance, a function grows exponentially is it is in the equivalence class of the∈ exponential). The growth rate type of contact homology is the growth rate of T rank(ϕT ). Two nondegenerate contact forms associated to the same contact structure7→ have the same growth rate type (by theorem 4.2) and therefore, the growth rate type of contact homology is an invariant of the contact structure. The growth rate of contact homology was introduced in [13]. It “describes” the asymptotic behavior with respect to T of the number of Reeb periodic orbits with period smaller than T that contribute to contact homology. For a more detailed presentation one can refer to [54]. Colin and Honda’s conjecture [18, Conjecture 2.10] (see section 1) for the contact structures from theorem 2.2, and theorem 2.18 for nondegenerate contact forms follow from

Proposition 4.4. Let (M, ξ) be a compact contact 3-manifold and assume there exists a contact form α0 on (M, ξ) whose Reeb ﬂow is Anosov with orientable stable and unstable foliations. Then any Rα0 -periodic orbit is even and hyperbolic.

Indeed, by theorem 3.4, one can apply theorem 4.4 to (MS, αA). Note that αA is hypertight as the Reeb ﬂow is Anosov. Therefore, the diﬀerential in contact homology is trivial (theorem 4.1.2.) and for any set Λ of free homotopy classes,

Λ M CHcyl(αA) = Qγ.

RαA -Reeb periodic orbit γ in Λ

Let α = fαA be nondegenerate with f > 0 and let B be such that 1/B f(m) B Λ ≤Λ ≤ for all m M. Applying theorem 4.3 for Λ = ρ , we get N (α) N (αA) ∈ { } L ≥ L/C(B) for any L > 0. Using the Barthelmé–Fenley estimates from [10, Theorem F] we obtain the desired logarithmic growth. This finishes the proof of theorem 2.18 in the nondegenerate case. Additionally, the number of periodic orbits of an Anosov flow in primitive homotopy classes grows exponentially with the period. Applying theorem 4.3 for Λ the set of all primitive free homotopy classes in MS proves the Colin–Honda conjecture for contact structures from theorem 2.2 and nondegenerate contact forms.

T Proof of theorem 4.4. By deﬁnition of stable and unstable foliations, Dϕ|ξ(p) has 1 + − real eigenvalues µ and µ and the associated eigenspaces are E and E . As the strong stable foliation is orientable, the eigenvalues are positive. Thus γ is even and hyperbolic

5. Orbit growth in a free homotopy class for degenerate contact forms In this Section, we prove theorem 2.18 for degenerate contact form (the nondegenerate case is explained in the previous section). The proof derives from the proof of [3, Theorem 1]. Yet Alves’ goal was to obtain one orbit with bounded period in some free homotopy class and not control the number of orbit in this class, and the following result is not explicit in [3]. 20 PATRICK FOULON, BORIS HASSELBLATT, AND ANNE VAUGON

Corollary 5.1. Let (M, ξ) be a closed manifold and α0 an Anosov contact form on (M, ξ). Let ρ be a primitive free homotopy class of M such that

ρ M CH (α0) = Qγ = 0 . cyl 6 { } Rα0 -periodic orbit γ in ρ

Then, for any contact form α = fαα0 on (M, ξ) and for any Rα0 -periodic orbit of 0 0 period T , there exists an Rα-periodic orbit in ρ of period T with eT T ET ≤ ≤ where e = min fα and E = max fα . | | | | Proof of theorem 5.1. Fix 0 < < e. Without loss of generality, we may assume fα > 0. We follow Alves’ proof of Theorem 1 in [3] and consider α = fαα0 on (M, ξ) nondegenerate. For any R > 0, Alves constructs (Step 1) a symplectic cobordism R MS between (E + )α0 and (e )α0 which corresponds to the symplectization × − of α on [ R,R] MS, and a map − × ρ ρ ΨR : CH ((E + )α0) CH ((e )α0) cyl −→ cyl − ρ by counting holomorphic cylinders in the symplectic cobordism. As CHcyl(Cα0) is ρ canonically isomorphic to CHcyl(α0) for any C > 0, ΨR induces an endomorphism ρ of CHcyl(α0) and Alves proves this endomorphism is, in fact, the identity.

Let γ be a Rα0 -periodic orbit of period T . For any C > 0, it induces a RCα0 periodic orbit γC of period CT . As

ρ M CHcyl(α0) = Qγ,

Rα0 -Reeb periodic orbit γ in ρ

ΨR(γE+) = γe− and therefore, there exists a holomorphic cylinder between γE+ and γe−. Now as R tends to infinity (Step 2), SFT compactness (see [3]) shows that our family of cylinders breaks and a Rα-periodic orbit γ of period T appears in a intermediate level. By construction (e )T T (E + )T . Now, let − ≤ ≤ 0 tend to 0 and use the Arzelà-Ascoli Theorem to obtain a Rα periodic orbit γ with period T 0 such that eT T 0 ET . ≤ ≤ If α is degenerate (Step 4), there exists a sequence (αn)n∈N of nondegenerate contact forms converging to α and the Arzelà-AscoliTheorem can again be applied to obtain the desired periodic orbit.

Proof of theorem 2.18 for degenerate contact forms. As MS is hyperbolic, there are a1, b1, a2, b2 > 0 such that

1 ρ ln(T ) c2 NT (αA) a1 ln(T ) + c1 a2 − ≤ ≤ for all T > 0 [10, Theorem F]. Let (γn)n∈N be a sequence of RαA -periodic orbits in ρ of periods (Tn)n∈N such that

γ0 is a RαA -periodic orbit in ρ with minimal period; • E for all n 0, γn+1 is a RαA -periodic orbit in ρ with period Tn+1 > e Tn • and such≥ that there exists no periodic orbit of smaller period satisfying the same conditions. 0 0 By theorem 5.1, for any n 0, there exists a Rα-periodic orbit γn of period Tn 0 ≥ 0 0 such that eTn Tn ETn. Therefore, Tn ETn < eTn+1 Tn+1 for all n 0 ≤ 0 ≤ ρ ≤ ≤ ≥ and all the orbits γn are distinct. Thus, NT 0 (α) n + 1 for all n 0. ρ n ≥ ≥ To control NT (α), we now estimate the growth of (Tn)n∈N. By deﬁnition, for all n 0, ≥ n ρ ρ o Tn+1 = min T N (α0) N (α0) + 1 . | T ≥ E/eTn ORBIT GROWTH OF CONTACT STRUCTURES AFTER SURGERY 21

Therefore, if T is such that 1 ln(T ) c2 = a1 ln(E/eTn) + c1 + 1 a2 − then Tn+1 T and ≤ E a1a2 (1+c1+c2)a2 Tn+1 Tn e . ≤ e a3 Therefore, there exist a3, c3 > 1 such that Tn+1 (c3Tn) for all n 0. Thus, ≤ ≥ there exists c4 > 0 such that n+1 ln(Tn+1) c4a ≤ 3 for all n 0 and there exists c5 R such that ≥ ∈ ln(ln(Tn+1)) ln(a3)(n + 1) + c5 ≤ 0 0 for all n 0. Now, if eTn−1 T T T ETn, then ≥ ≤ n−1 ≤ ≤ n ≤ ρ 1 1 NT (α) n ln(ln(Tn)) c5 ln(ln(T )) c6 ≥ ≥ ln(a3) − ≥ ln(a3) − for some c6 R. This proves theorem 2.18. ∈ Remark 5.2. If a1a2 = 1, one can get better estimates and obtain the same growth as in the nondegenerate case.

6. Exponential growth of periodic orbits after surgery on a simple geodesic We now prove theorem 2.22 using the following result by Alves. To state it, we ﬁrst deﬁne the exponential homotopical growth of cylindrical contact homology. Let (M, ξ) be a closed contact manifold and α0 a hypertight contact form on (M, ξ). cyl For T > 0, let NT (α0) be the number of free homotopy classes ρ of M such that

all the Rα0 -periodic orbits in ρ are simply-covered, nondegenerate and have • period smaller than T ; ρ CH (α0) = 0. • cyl 6 Deﬁnition 6.1 (Alves [1]). The cylindrical contact homology of (M, α0) has exponential homotopical growth if there exist T0 0, a > 0 and b R such that, for ≥ ∈ all T T0, ≥ cyl aT +b N (α0) e . T ≥ Theorem 6.2 (Alves [1], Theorem 2). Let α0 be a hypertight contact form on a closed contact manifold (M, ξ) and assume that the cylindrical contact homology has exponential homotopical growth. Then every Reeb ﬂow on (M, ξ) has positive topological entropy.

If ρ is a free homotopy class containing only one Rα0 -periodic orbit and if this orbit is simply-covered and nondegenerate, it is a direct consequence of the deﬁni- ρ tion of contact homology that CHcyl(α0) = Q. Therefore, to prove theorem 2.22, it suﬃces to prove the following propositions.

Proposition 6.3. The contact form αA is hypertight in MS.

Proposition 6.4. Let (MS, αA) be a contact manifold obtained after a contact 0 surgery along a simple geodesic. Let NT (αA) be the number of free homotopy classes

ρ such that ρ contains only one RαA -periodic orbit and this orbit is simply-covered, nondegenerate and of period smaller than T . Then, there exist T0 0, a > 0 and 0 aT +b ≥ b R such that, for all T T0, NT (αA) e . ∈ ≥ ≥ 22 PATRICK FOULON, BORIS HASSELBLATT, AND ANNE VAUGON

0 Indeed, the exponential growth of NT (αA) with respect to T induces the exponential homotopical growth of (MS, αA) and we can apply theorem 6.2. We now turn to the proofs of theorem 6.3 and theorem 6.4. In SΣ, T = π−1(c) is a torus, and our surgery preserves this torus. Let TS denote the associated torus in MS. Van Kampen’s Theorem tells us that TS is π1-injective. To prove theorem 6.4, we want to ﬁnd free homotopy classes with only one periodic Reeb orbit. We will consider free homotopy classes containing a periodic orbit disjoint from TS and prove there are enough of such classes. First, we describe Reeb periodic orbits and study the properties of free homotopies between them.

Claim 6.5. There are three types of RαA -periodic orbits: (1) periodic orbits contained in TS, the only periodic orbits of this kind are c, c (c with the reverse orientation) and their covers, − (2) periodic orbits disjoint from TS, these orbits correspond to closed geodesics in Σ disjoint from π(c) (this includes multiply-covered geodesics), (3) periodic orbits intersecting TS transversely.

Therefore, a free homotopy between two RαA -periodic orbits can always be per- turbed to be transverse to TS. 1 Proposition 6.6. Let δ0, δ1 be two smooth loops in MS and H : [0, 1] S MS −1 × → be a free homotopy between δ0 and δ1 transverse to TS. N := H (TS) is a smooth manifold of dimension 1 properly embedded in [0, 1] S1. Therefore, × (1) one can modify H so that N does not contain contractible circles,

(2) if δ0 is a RαA -periodic orbit transverse to TS, N does not contain a segment with end-points on 0 S1. { } × 1 Proof. Consider an innermost contractible circle c0 in N [0, 1] S , c0 bounds a 1 ⊂ × disk D0 in [0, 1] S . The image of c0 is contractible in TS as TS is π1-injective. × Therefore, there exists a continuous G: D0 TS such that H|c0 = G|c0 and one can → replace H|D0 by G to obtain a new homotopy (still denoted by H ) between δ0 and δ1. Now, consider a neighborhood [ ν, ν] TS of TS in MS (with TS 0 TS) − × '{ } × and a disk D1 containing D0 such that H(D1) [0, ν] TS and H(D1 r D0) ⊂ × ⊂ (0, ν] TS. One can perturb H in int(D1) so that H(D1) (0, ν] TS. Performing this inductively× on the contractible circles proves 1. ⊂ ×

We now assume δ0 is an RαA -periodic orbit transverse to TS. By contradiction, 1 consider an innermost segment c0 in N with end-points on 0 S . The end-points { }× of c0 correspond to consecutive intersection points of δ0 with TS. Let c1 be the segment in 0 S1 joining these two end-points points and homotopic (relative to { } × end-points) to c0. By construction, there exists a homotopy (ηt) : [0, 1] MS t∈[0,1] → (relative to end-points) between η0 = H(c0) et η1 = H(c1) such that ηt(s) TS if and only if t = 1 or s = 0, 1. Let M 0 be the manifold with boundary obtained∈ 0 by cutting MS along TS. Note that M can also be obtained by cutting SΣ along 0 0 T. The projection M MS is injective in the interior of M , therefore ηt(s) is well-deﬁned in M 0 if t =→ 0 and s = 0, 1. Thus, there exists a homotopy η0 in M 0 6 6 t lifting ηt. This homotopy induces a homotopy in SΣ and, as a result, a homotopy in Σ between a geodesic arc contained in π(c) and a geodesic arc with end-points on π(c). As Σ is hyperbolic, this can only happen if our second geodesic arc is also contained in π(c), a contradiction. Proof of theorem 6.3. By contradiction, assume there exists a free homotopy H between δ, a RαA periodic orbit, and a point p / TS. As TS is π1-injective, δ ∈ cannot be contained in TS. Without loss of generality we may assume that H is transverse to TS and apply theorem 6.6. ORBIT GROWTH OF CONTACT STRUCTURES AFTER SURGERY 23

1 If δ is disjoint from TS, then N [0, 1] S (see theorem 6.6) can only contain circles parallel to the boundary. We⊂ will now× prove that we can modify H so that 1 N is empty. Let c0 be the circle in N closest to 0 S and let C be the closure of 1 { }× 1 the connected component of ([0, 1] S )rc0 containing 1 S . Then H(c0) is an × { }× immersed circle contractible in TS and there exists a continuous map G: C TS → 1 such that G|c0 = H and G{1}×S is constant. We replace H|C with G to obtain a new homotopy H. Now, consider a neighborhood [ ν, ν] TS of TS in MS and a − × neighborhood C1 of C such that H(C1) is contained in [0, ν] TS. We can perturb × H so that H(C1) is contained in (0, ν] TS. Therefore we may assume that N × is empty and H is an homotopy in MS r TS. It induces an homotopy in SΣ, a contradiction as the periodic orbits are not contractible in SΣ. Finally, we consider the case δ transverse to TS. In this case, N has boundary points on 0 TS but not on 1 TS. This contradicts theorem 6.6. { } × { } ×

Proposition 6.7. If δ is a RαA -periodic orbit disjoint from TS, then the free homotopy class of δ contains exactly one RαA -periodic orbit.

Proof. By contradiction, consider a free homotopy H from δ to δ1, a distinct RαA - periodic orbit. Without loss of generality, we may assume that H is transverse to TS apply theorem 6.6 If δ1 is disjoint from TS, then N can only contain circles parallel to the boundary. If N is empty, H induces a homotopy in SΣ and therefore in Σ. Yet, two closed geodesics on a hyperbolic surface are not homotopic. This proves N is not empty. 1 0 Let c0 be the circle in N closest to 0 S and M be the manifold with boundary { }× obtained by cutting MS along TS. The homotopy H induces a homotopy G between 0 δ and H(c0) TS. The homotopy G lifts to M and therefore induces a free homotopy in S⊂Σ and, as a result, a free homotopy in Σ between a closed geodesics and a loop contained in the geodesic π(c). This can happen only if our first geodesic is a cover of π(c). Yet this implies δ TS, a contradiction. ⊂ If δ1 is transverse to TS, the manifold N is not empty and has end-points on 1 S1 but cannot have end-points on 0 S1. This contradicts theorem 6.6. { } × { } × Finally, the case δ1 contained in TS is similar to the case δ1 disjoint from TS. In this case, N contains only circles parallel to the boundary and 1 S1 is in { } × N. Proof of theorem 6.4. If π(c) is nonseparating, by cutting Σ along π(c) we obtain a surface of genus at least 1 with two boundary components. Let `1 and `2 be two loops in Σ r c homotopically independent and with the same base-point. Then, any nontrivial word in `1 and `2 defines a nontrivial free homotopy class for Σ and there exists a closed geodesic on Σ representing this class. This RαA -periodic orbit is always nondegenerate. Additionally, we may assume that the orbits associated to `1 and `2 are simply-covered. If a word is not the repetition a smaller word, the associated orbit is therefore simply covered. As `1 and `2 are independent all these geodesics are disjoint and their number grows exponentially with the period. Finally, these geodesics do not intersect c as geodesics always minimize the intersection number. If π(c) is separating, by cutting Σ along π(c) we obtain two surfaces of genus at least 1 with one boundary components. The proof is similar. 7. Coexistence of diverse contact flows—proof of theorem 2.23 7.1. Dynamical properties of the periodic Reeb flow after surgery. We now apply the general construction of contact surgery along a Legendrian curve described in section 3.1 to the contact structure with contact form β and periodic Reeb flow described in section 2.4.3. On the unit tangent bundle of a hyperbolic 24 PATRICK FOULON, BORIS HASSELBLATT, AND ANNE VAUGON surface Σ, select a closed geodesic c: S1 = R/Z Σ, s c(s) and consider the Legendrian knot γ obtained by rotating the unit→ vector7→ field along c by the angle θ = π/2. Note that the Legendrian knot γ is the same as in section 3.2 (and is tangent to H). To obtain standard coordinates in a neighborhood of γ we first consider an annulus A in SΣ transverse to the fibers with coordinates 1 (s, w) S ( 2, 2) such that β|A = w ds and then flow along the Reeb vector ∈ × − 1 7 field Rβ to obtain coordinates (t, s, w) S A = N such that β = dt + w ds (to remain coherent with previous conventions∈ × our circles have different lengths, more precisely t R/2πZ and s R/Z). Note that N can be interpreted as the suspension of the∈ annulus A by the∈ identity map. Our non-trivial surgery is defined by a twist (shear) F along A. We denote by MS the manifold SΣ after surgery and by NS MS the manifold (with boundary) ⊂ N after surgery. Let βS be the contact form on MS as described in section 3.1. Note that β and βS coincide outside N and NS respectively. The manifold NS is the suspension of the annulus A by the shear map F . Moreover, the map pS : NS ( 2, 2) given by the w-coordinate is well-defined and is a trivial torus-bundle. For→ w− ( 2, 2), the torus p−1(w) is foliated by closed Reeb orbits if and only if ∈ − S pw f (w) = 2π 2πQ qw ∈ ∂ −1 where pw and qw are coprime. In this situation the orbits of ∂t on pS (w) are ∂ periodic of period qw. The Reeb vector field is a renormalization of ∂t (see (7)). 1 1 Finally, let T = S S 0 in N and TS be its image in MS. By van Kampen’s × × { } theorem, this torus is incompressible. Therefore the contact form βS is hypertight. Note that if f(w) 2πQ and f(w0) 2πQ but f(w) = f(w0), the associated periodic orbits are not∈ freely homotopic.∈ 6

7.2. Proof of theorem 2.23. The contact form βS is degenerate and the renormalization from the surgery makes the direct study a bit harder. So, to estimate the growth rate of its contact homology, we will standardize and perturb our contact form. ∂ f(w) ∂ ∂ For any w ( 2, 2), the vector fields ∂t + 2π ∂s and ∂s generate circles in −1 ∈ − the torus pS (w). These circles induce a trivialisation of NS. Let (τ, σ, w) be the coordinates on NS associated to this trivialisation. Without loss of generality, we may assume that the map f defining the twist (shear) F is constant on ( 2, ) (, 2), that f 0(w) = 0 for any w ( , ) and that f is invariant under− reflection− ∪ with respect to the6 point (0, q/2).∈ Therefore,− for w in [ 2, ], − − βS = dτ + w dσ and for w in [, 2], qw β = 1 + dτ + w dσ. S 2π

Lemma 7.1. There exist smooth maps h0, k0 :( 2, 2) R such that − → β0 = h0(w) dτ + k0(w) dσ

7These coordinates along γ are different from the coordinates defined for the surgery associated to the contact form α as, for instance, the surgery annulus is different. It is possible to derive a contact form from β on the surgered manifold using the coordinates and surgery associated to α: write β in local coordinates, compute F ∗β and interpolate using bump and cut-off functions. Unfortunately, this construction yields a complicated Reeb vector field. Note that the contact structure obtained this way is isotopic to ker(βS ). This can be proved as follows. First the two surgeries result in the same manifold. Moreover, a surgery can be described as the gluing of a solid torus on an excavated manifold. Therefore we just need to prove that the contact structures 3 on the glued tori are the same. This derives from the classification of contact structures on D by Eliashberg. See [42] for an application to the torus. ORBIT GROWTH OF CONTACT STRUCTURES AFTER SURGERY 25 is a contact form such that β0 = βS for w close to 2 and Rβ and Rβ are ± 0 S positively collinear on NS. Therefore, β0 and βS are isotopic (through contact forms).

Proof. Let h0 and k0 be the maps defined by k0(w) = w and Z w h0(w) = 1 + f(u)/2π du −2 R for w ( 2, 2). As − f(u)du = q, β0 = βS for w (, 2). Moreover, the contact∈ condition− is ∈ Z w 1 + f(u)/2π du wf(w)/2π > 0 −2 − and this condition is always satisfied for small enough. Additionally, the Reeb vector field is positively collinear to (k0 (w), h0 (w)) = (1, f(w)/2π). Finally, as 0 − 0 − Rβ and Rβ are positively collinear, (uβS +(1 u)β0) is a contact isotopy. 0 S − u∈[0,1]

The contact form β0 is degenerate. To estimate the growth rate of its contact homology, we have to perturb it. Our perturbation draws its inspiration from Morse–Bott techniques. To describe our perturbation, we need to fix some notations. The manifold SΣ r p−1(( , )) is a trivial circle bundle. Let S0 be a surface (with boundary) transverse− to the fibers and intersecting each fiber once: S0 provides us with a trivialisation S0 S1 of SΣ r p−1(( , )). The surface S0 × − has two boundary components. Let ϕ: S0 R be a Morse function such that ϕ = 0 on the boundary of S0 and, if q > 0→ (resp. q < 0), the connected component of ∂S0 corresponding to w = is a maximum (resp. a minimum) and the connected component corresponding− to w = a minimum (resp. a maximum). For any w such that f(w) = 2πp(w)/q(w) 2πQ, we denote by P (w) the period of −1 ∈ the Rβ0 -periodic orbits foliating pS (w). Note that there exists CP > 0 such that q(w)/CP P (w) CP q(w), this implies that the number of torus with w ( , ) foliated by≤ Reeb periodic≤ orbits with period smaller than L grows quadratically∈ − in L. For a contact form α, let σ(α) denote the action spectrum: the set of periods of the periodic orbits of Rα. 0 Proposition 7.2. Let T > 0, T/ σ(β0). There exists β = lβ0 with l : MS R+ arbitrarily close to 1 such that ∈ → β0 is hypertight and nondegenerate • the periodic orbits of Rβ0 with period T are exactly: • (1) the fibers associated to the critical≤ points of ϕ and their multiple of multiplicity T ≤ 2π (2) for all w ( , ) such that P (w) < T , two orbits in p−1(w) and their ∈ − S multiple with multiplicity T ≤ P (w) if δ is a Rβ0 -periodic orbit of period T then all the Rβ0 -periodic orbit in • the free homotopy class of δ are periodic≤ orbits of period T . ≤ Proposition 7.3. If δ is a simply-covered Rβ0 -periodic orbit of period T of the second type in theorem 7.2, then ≤

[δ] 2 CHcyl(M, ker(β0)) = Q . Proof of theorem 7.2. There exists ν > 0 such that for any w ( , + ν] [ ∈ − − ∪ − ν, ), if f(w) = 2πp(w)/q(w) 2πQ then q(w) > CP T . Let ∈ N 0 = p−1(( , + ν] [ ν, )). S S − − ∪ − 26 PATRICK FOULON, BORIS HASSELBLATT, AND ANNE VAUGON

00 0 Let S be a smooth surface in MS with boundary obtained by adding to S two annuli in NS, transverse to Rβ and projecting to [ , + ν] [ ν, ]. We can 0 − − ∪ − therefore endow S00 r S0 with coordinates (s0, w0) such that w0 lifts w. We now perturb ϕ and extend it to S00 so that ϕ(s0, w0) = ϕ(w0) on S00 r S0, ϕ0(w0) = 0 for all w0 [ , + ν) ( ν, ], ϕ is ﬂat (all its derivative are equal to 0)6 for w = ( ∈ν−) and− the critical∪ − points of ϕ are unaltered. Finally, we extend ϕ to ± − 0 MS to obtain a smooth function, Rβ0 -invariant and such that ϕ 0 in NS r NS. ≡ Let βλ = (1 + λϕ)β0. This is a standard Morse–Bott perturbation (see [12, −1 Lemma 2.3]) in MS r pS (( , )), therefore, for λ 1, the periodic orbits in this area correspond to the critical− points of k. In the coordinates (τ, σ, w), we have

βλ = (1 + λϕ(w))(h0(w) dτ + k0(w) dσ). Therefore, in these coordinates, the Reeb vector ﬁeld is positively collinear to

0 0 ∂ 0 0 ∂ ((1 + λϕ(w))k (w) + λϕ (w)k0(w)) ((1 + λϕ(w))h (w) + λϕ (w)h0(w)) . 0 ∂τ − 0 ∂σ The σ-coordinate is nonzero as ϕ and h have the same monotonicity. For λ 1, 0 0 the σ-coordinate is close to h0(w), the τ-coordinate to k0(w) and Rβλ is close to − 0 Rβ0 . Therefore, for λ 1, if there is a Rβλ -periodic orbit in NS, this orbit has 0 0 0 slope 2πp (w)/q (w) 2πQ with q (w) > CP T . Thus there are no periodic orbit ∈ 0 with period smaller than T in NS and the periodic orbits with period bigger than T are not in the free homotopy classes of orbits with period smaller than T as described in theorem 7.2. −1 In pS ([ + ν, ν]), the periodic orbits with period T are contained in tori −1 − − ≤ pS (w) such that P (w) T . These tori are foliated by periodic orbits. Morse– Bott techniques apply here≤ and give the second type of periodic Reeb orbits: for −1 any such w we perturb β in a neighborhood of pS (w) with a function derived −1 1 from a Morse function ϕw defined on pS (w)/Reeb flow = S and the periodic orbits after perturbation correspond to the critical points of ϕw. For a given w we obtain two orbits (one associated to the maximum of ϕw and one associated to the minimum of ϕw), their covers and some orbits with period bigger than T and in the free homotopy class of arbitrarily large covers of our two simple orbits. This perturbation derives from [12, Lemma 2.3] and is described for tori in [54, Section 3.1]. Lastly, standard perturbation techniques prove there exists an arbitrarily small perturbation of βλ with the following properties: it gives rise to a nondegenerate contact form, • it does not change the periodic orbits with period smaller than T , • it does not create periodic orbits of period bigger than T in the free homo- • topy classes of orbits of period smaller than T . −1 Proof of theorem 7.3. Let δ p (w) be a Rβ0 -periodic orbit of period T of ∈ S ≤ the second type in theorem 7.2. Then the Rβ0 -periodic orbit in the class [δ] are −1 exactly the orbits in pS (w) (and all these orbits have the same period). As δ is simply-covered, Dragnev’s [20] results can be applied. Additionally, standard perturbations do not create contractible periodic Reeb orbits. Therefore, the differential for contact homology can be described using “cascades” from Bourgeois’ work [12]. The case of a unique torus of orbit is explained in [12, Section 9.4]. The cascades used to describe the differential in this degenerate setting mix holomor- −1 1 phic cylinders between orbits and gradient lines for ϕw in pS (w)/Reeb flow = S (for some generic metric). As all periodic orbits in this class have the same period, there is no homolorphic cylinder in the cascade and the differential coincides ORBIT GROWTH OF CONTACT STRUCTURES AFTER SURGERY 27 with the Morse–Witten differential for ϕw (ie the differential associated to Morse homology). Therefore, cylindrical contact homology in the free homotopy class ρ is 2-dimensional. The cascades of Morse–Bott homology are explicitly described in [15] (in a slightly different setting). Proof of theorem 2.23. Let β0 = fβ be a nondegenerate hypertight contact form and B be such that 1/B < f < B. Let (Ti)i∈N be an increasing sequence such that limi→+∞ Ti = + and Ti / σ(β0). For all i N, let βi = fiβ be the contact form ∞ ∈ ∈ given by theorem 7.2 for T = Ti. We may assume, 1 f < i < B B f as fi is arbitrarily close to 1. By theorem 7.2, Ti X Ti dim (CHTi (αi)) C + 2 ≤ 2π P (w) w,P (w)≤Ti where C is the number of critical points of k and X T = O(T 2). P (w) i w,P (w)≤T In addition, we have the following commutative diagram (see theorem 4.1),

0 CH≤Ti/C(B)(β ) CH≤Ti (βi)

0 ϕTi/C(B)(β ) ϕTi (βi)

0 CH(β ) CH(βi) thus 0 2 rk(ϕTi/C(B)(β )) rk(ϕTi (βi)) dim (CHTi (βi)) a1(Ti ). ≤ ≤ ≤ A symmetric commutative diagram implies 0 rk(ϕ (βi)) rk(ϕT (β )) Ti/C(B) ≤ i

Propositions 7.2 and 7.3 prove that ϕTi/C(B) is injective on the class of simply- covered periodic orbits of the second type (as defined in theorem 7.2). Therefore 2 rk(ϕ (βi)) a0T and the growth rate of contact homology is quadratic. Ti/C(B) ≥ i References [1] Marcelo Ribeiro de Resende Alves: Cylindrical contact homology and topological entropy. Geometry & Topology 20 (2016), no. 6, 3519–3569 [2] Marcelo Ribeiro de Resende Alves: Legendrian contact homology and topological entropy. arXiv:1410.3381 [3] Marcelo Ribeiro de Resende Alves: Positive topological entropy for Reeb flows on 3- dimensional Anosov contact manifolds. Journal of Modern Dynamics 10 (2016), 497–509 [4] Marcelo Ribeiro de Resende Alves, Vincent Colin, and Ko Honda: Topological entropy for Reeb vector fields in dimension three via open book decompositions. arXiv:1705.08134 [5] Thierry Barbot: Caractérisationdes flots d’Anosov en dimension 3 par leurs feuilletages faibles. Ergodic Theory Dynam. Systems 15 (1995), no. 2, 247–270. [6] Thierry Barbot: De l’hyperbolique au globalement hyperbolique, Habilitation àdiriger des recherches, UniversitéClaude Bernard de Lyon. http://www.univ-avignon.fr/fileadmin/documents/Users/Fiches_X_P/memoireCRY.pdf [7] Luis Barreira, Yakov Pesin: Nonuniform hyperbolicity. Dynamics of systems with nonzero Lyapunov exponents. Encyclopedia of Mathematics and its Applications, 115. Cambridge University Press, Cambridge, 2007 [8] Thomas Barthelmé: A new Laplace operator in Finsler geometry and periodic orbits of Anosov flows. Doctoral thesis, Universitéde Strasbourg, 2012, arXiv:1204.0879 28 PATRICK FOULON, BORIS HASSELBLATT, AND ANNE VAUGON

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Centre International de Rencontres Mathematiques,´ UMS 822, and Institut de Mathematiques´ de Marseille, UMR 7373, 13453 Marseille, France E-mail address: [email protected]

Department of Mathematics,Tufts University, Medford, MA 02155, USA E-mail address: [email protected]

Laboratoire de Mathematiques´ d’Orsay, Univ. Paris-Sud, CNRS, Universite´ Paris- Saclay, 91405 Orsay, France E-mail address: [email protected]